Some results about equichordal convex bodies

TL;DR

This paper investigates the existence of equichordal bodies within convex bodies, showing that in higher dimensions only spheres have such bodies, and explores related geometric properties like isoptic curves.

Contribution

It proves the existence of non-circular equichordal bodies in planar convex bodies and establishes that only spheres have them in dimensions three and higher.

Findings

Multiple non-circular equichordal convex bodies exist in the plane.

In dimensions ≥3, only Euclidean balls have equichordal convex bodies.

Connections between isoptic curves and convex rotors are established.

Abstract

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 2$, with $L\subset \text{int}\, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $\lambda$. J. Barker and D. Larman proved that if $L$ is a ball, then $K$ is a ball concentric with $L$. In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. We also prove some results about isoptic curves and give relations between isoptic curves and convex rotors in the plane.